Differential-algebraic systems are generically controllable and stabilizable

We investigate genericity of various controllability and stabilizability concepts of linear, time-invariant differential-algebraic systems. Based on well-known algebraic characterizations of these concepts (see the survey article by Berger and Reis (in: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations I, Differential-Algebraic Equations Forum, Springer, Berlin, pp 1–61. https://doi.org/10.1007/978-3-642-34928-7_1)), we use tools from algebraic geometry to characterize genericity of controllability and stabilizability in terms of matrix formats.


R k×
The vector space of all k × matrices with entries in a ring R. rk F M The rank of M ∈ R k× over the field F. [Rx 1 , . . . , x n ] := k=0 a k x

Introduction
We study genericity of controllability and stabilizability of differential algebraic systems described by the equation where (E, A, B) ∈ ,n,m := R ×n × R ×n × R ×m .
To be precise, we first say what we understand under genericity.
Definition 1.1 [13, p. 28] and [11, p. 50] A set V ⊆ R n is called an algebraic variety, if there exist finitely many polynomials such that V is the locus of their zeros, i.e., An algebraic variety V is called proper if V R n . The set of all algebraic varieties in R n is denoted as A set S ⊆ R n is called generic, if there exists a proper algebraic variety V ∈ V prop n (R) so that S c ⊆ V. If the algebraic variety V is known, then we call S generic with respect to (w.r.t.) V.
"Generic" is not consistently used in the literature. We show in the following proposition that generic as in Definition 1.1 is stronger than containing an open and dense subset with respect to the Euclidean topology. Proof (i) We show that V is closed. Let V be given as in (2). Then, each p −1 i ({0}) is closed since p i is continuous and hence, the claim follows.

Proposition 1.2 Any set S ⊆ R n satisfies:
It remains to prove that V c is dense. Seeking a contradiction, suppose that V c is not dense or, equivalently, V has at least one inner point. Then, λ n (V) > 0 and V cannot be a proper algebraic variety by Proposition A.3.
(ii) This is an immediate consequence of (i). Now, Lemma A.7 shows that S is not generic. If S c = V were an algebraic variety, then S is generic w.r.t. V which is a contradiction. This completes the proof.
To characterize genericity in terms of the Zariski topology, recall [11, p. 50] that the latter is defined by the property that all closed sets are the algebraic varieties. The Zariski topology is strictly coarser than the Euclidean topology and we have:

is generic if, and only if, S contains a nonempty Zariski open set.
This approach was used by Belur and Shankar in their investigations of genericity of impulse controllable systems (see [3,Section 3]). Since they consider differentialalgebraic equations described by differential operator matrices and hence an infinite dimensional vector space, they need to extend the definition of generic sets to this space using the limit topology of the Zariski topology. This is not necessary in our setup.
In the special case that (1) is an ordinary differential equation, that is = n and E = I , Lee and Markus [10] proved that that the set of all controllable systems is open and dense w.r.t. the Euclidean topology. Wonham [12,Thm. 1.3] showed in the first edition of his monograph that the set of all controllable systems is generic.
Recently, it has been shown that linear, time-invariant port-Hamiltonian systems are generically controllable; see [9].
When it comes to differential-algebraic equations, then to the best of our knowledge there are only two contributions known where open and dense subsets of controllable systems are investigated. Banaszuk and Przyłuski [1] consider an algebraic criterionwhich they do not justify analytically and which is not related to any concept of controllability-and give a sufficient condition so that the set of systems satisfying the algebraic criterion contains an open and dense subset. The second contribution is by Belur and Shankar [3].
Their main interest is on polynomial systems, and if specialized to matrix pencils, they derive a characterization of genericity of impulse controllability. Other concepts are not studied.
The basis of our approach is the algebraic characterizations of various concepts of controllability and stabilizability of differential-algebraic equations; this is well known and summarized in Propositions 2.1 and 3.1. We characterize-in terms of the formats , n, and m of (1)-when these controllability and stabilizability concept hold generically. This is the content of Theorems 2.3 and 3.3. The proofs of these two main results are based on methods from algebraic geometry, tailored for our purposes and relegated to "Appendix 1", and some results on ranks of special matrices are presented in "Appendix 2".

Controllability
There are various controllability concepts for differential-algebraic equations (1) such as freely initializable (also called 'controllable at infinity'), impulse controllable, completely controllable, behavioral controllable, and strongly controllable. Their definitions and their algebraic characterizations are given in the next proposition. To state this, we need to say what a solution of (1) is. We consider the behavior of (1) given by Note that any f ∈ W 1,1 loc (R, R d ) is continuous. Controllability of a system (1) is a property of the corresponding behavior B [E,A,B] . If systems described by ordinary differential equations are considered, i.e., the special case E = I , then the initial value x 0 ∈ R n can be freely chosen and the problem is to which other points it can be steered in finite time. It is well-known that the system (1) with E = I is called controllable if, and only if, for any given initial state x 0 ∈ R n and any terminal state x 1 ∈ R n , there exists a control u ∈ L 1 loc (R, R m ) which steers x 0 to x 1 in finite time T > 0, more formally, there exists (x, u) ∈ B [E, A,B] such that x(0) = x 0 and x(T ) = x 1 . However, if an arbitrary matrix E is allowed in (1), then algebraic constraints are added to the differential equation. So it is unclear as to whether the initial value can be chosen freely. If the latter is the case, then the system is called freely initializable, sometimes also called controllable at infinity.
If (1) is assumed to be freely initializable, then one may ask, whether each initial state can be steered to any final state in finite time. If both conditions are fulfilled, then the system is called completely controllable.
A stronger controllability concept-but also a generalization of the ODE case-is in the behavioral setup the problem as to whether it is always possible to concatenate two given solutions ( The concepts of freely initializable and completely controllable systems can be weakened in the sense that the initial and the terminal value are compared with respect to the image of E; for example, the initial condition becomes E x 0 = E x(0). These weakened concepts are called impulse controllable and strongly controllable.
The precise definitions and algebraic characterizations are given in the following proposition.

Proposition 2.1 For any (E, A, B)
∈ ,n,m , the following controllability definitions associated with the system (1) are algebraic characterized as follows: : : Proof Berger and Reis [4] derive a feedback form and use this as a tool in conjunction with 'canonical' representatives of certain equivalence classes to prove all characterizations of controllability in their survey. Note that in their characterization of strongly controllability, the term '+im R E' is missing in the first respective line in [4,Cor. 4.3]. Genericity of the different controllability concepts can be characterized in terms of the system dimensions. To this end, we introduce the notation where 'controllable' stands for one of the controllability concepts.

Theorem 2.3
For each of the controllability concepts defined in Proposition 2.1, the following characterizations hold: Proof We proceed in steps.
Step 1 We show: S freely initial. is generic if, and only if, ≤ n + m. First note that Proposition 2.1 yields and therefore S (i) ∩ S (ii) ⊆ S c freely initial. and S c freely initial. is generic. Thus, Lemma A.6 shows that S freely initial. is not generic.
⇐ Since ≤ n + m , the sets are both non-empty and by Proposition B.3 (i) and (ii) they are generic. Now, Corollary A.5(ii) yields that S (i) ∩ S (ii) is a generic set, and by Remark A.1, S freely initial. ⊇ S 1 ∩ S 2 is generic, too.
Step 2 We show: S imp. contr. is generic if, and only if, ≤ n + m. We consider the two cases ≥ n and < n. ≥ n: By Proposition B.3 (i) and (iii), the sets hold true. By Proposition 2.1, we find that S ∩ S (i)] ⊆ S imp. contr. and hence, in view of Corollary A.5(ii) and Remark A.1, the set S imp. contr. is generic.
Step 3 Proposition 2.1 yields We show: S beh. contr. is generic if, and only if, = n + m. From Proposition B.5, we find that the set is a generic set. The equation (5)  In the following remark, we stress the observation that linear differential-algebraic systems are either generically controllable or generically not controllable.

Remark 2.4 A closer inspection of the proof of Theorem 2.3 yields that if S arbitrary controllability
is not generic, then it is contained in a proper algebraic variety and thus its complement is generic.

Stabilizability
In the present section, genericity of stabilizability of DAEs is studied.
In the ODE-case, a system (1) is called stabilizable if, and only if, for each initial value x 0 ∈ R n there exists a control u ∈ L 1 loc (R, R m ) which steers the forced trajectory x in (possibly) infinite time to zero, that is (x, u) ∈ B [I ,A,B] so that x(0) = x 0 and lim t→∞ ess sup x|(t, ∞)| = 0.
As for controllability, different generalizing concepts have to be studied for DAEs. Each system that is stabilizable in the ODE-sense is called completely stabilizable. Similar to controllability, this concept is weakened if only E x(t) is considered-in this case we speak of strong stabilizability. Finally, a system is called behavioral controllable if, and only if, each (x, u) ∈ B [E,A,B] can be concatenated with some (x, u) ∈ B [E,A,B] which tends to zero as t tends to infinity.
The precise definitions and algebraic characterizations are given in the following proposition. We write W 1,1 loc (I , R n ) for the set of all weakly differentiable ϕ ∈ L 1 loc (R, R n ). A, B) ∈ ,n,m , the following controllability definitions associated with the system (1) are algebraic characterized as follows:
beh. stabl.  where 'stabilizable' stands for one of the stability concepts. yields that S str. stabl. is generic if, and only if < n + m.

Theorem 3.3 For each of the three stabilizability concepts from Proposition
As for controllability, we would like to emphasize that linear differential-algebraic systems are either generically controllable or generically not controllable.

Remark 3.4 A closer inspection of the proof of Theorem 3.3 yields as in Remark 2.4 that S stabilizable is either generic or contained in a proper algebraic variety.
Acknowledgements We are indebted to our colleague Thomas Hotz (Ilmenau) for several constructive discussions, and to a reviewer who read the two submissions very carefully and made many helpful comments.

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Appendix A Algebraic geometry
The results presented in the present section are tailored from algebraic geometry for our needs. They provide a basis for the proofs of Theorems 2.3 and 3.3.
Throughout this section, let n ∈ N * . We identify, wherever needed, any polynomial-an algebraic object- with the polynomial -an algebraic object - An immediate consequence of Definition 1.1 is the following useful property.
prop n (R), then any encompassing set S ⊇ S is generic w.r.t. V. Definition 1.1 can be simplified using one instead of finitely many polynomials.

Lemma A.2 Any set V ⊆ R n is an algebraic variety if, and only if, there is a polyno-
We use the well-known result that proper algebraic varieties are Lebesgue null sets.
Proof The intersection and union of algebraic varieties are algebraic varieties; see [11, p. 50]. Since the intersection and union of finitely many Lebesgue null sets are null sets, the claim follows from Proposition A.3.
By Definition 1.1 and Remark A.1, we have the following corollary. With the help of Corollary A.5, we conclude from Proposition A.3 that R n cannot be partitioned into more than one generic set.

Lemma A.6 R n cannot be partitioned into two generic sets.
Proof Seeking a contradiction, suppose S 1 , S 2 ⊆ R n is a partition of R n into generic sets, i.e., S 1 , S 2 are nonempty disjoint sets with S 1 ∪ S 2 = R n and S i is generic The latter is a proper algebraic variety by Corollary A.4, but this contradicts Proposition A.3.
Another consequence of Proposition A.3 is the following lemma, which will be used for the simplification of some proofs.
Lemma A.7 If λ n (S) < ∞ for some set S ⊆ R n , then S is not generic.
Proof If S ⊆ R n were a generic set with λ n (S) < ∞, then S c is by Corollary A.5 a Lebesgue null set and additivity of the Lebesgue measure yields the contradiction ∞ = λ n (R n ) = λ n (S ∪ S c ) = λ n (S) + λ n (S c ) = λ n (S) < ∞.
A useful property of generic sets is the next lemma.
Lemma A.8 Let S 1 ⊆ R n be generic and S 2 , S ⊆ R n with S ∩ S 1 = S 2 ∩ S 1 . Then, S is generic if, and only if, S 2 is generic.
Proof Since the statement is symmetric, only one direction has to be shown. Suppose S is generic. Then, Corollary A.5 yields that S ∩ S 1 = S 2 ∩ S 1 is generic. Since S 2 ∩ S 1 ⊆ S 2 , genericity of S 2 follows from Remark A.1.

Proposition B.3 The following sets are proper algebraic varieties
(i) (E, A, B is an algebraic variety, whence S c an algebraic variety by Corollary A.5 (ii). Since S is nonempty, S c is a proper algebraic variety.
The remaining proofs of (ii)-(iv) are similar and omitted.
In the remainder of this section, we investigate polynomial matrices of degree one, also called a pencil, of the form  (7) on ,n,m .
Note that r ≥ 1 since d ≤ min { , n + m}. Then, we see that the functions h i := ϕ • M i : ,n,m → R d+1 , defined for all i ∈ r , are well defined polynomial vectors and satisfy, in view of (8), the equivalence is an algebraic variety, whence S c an algebraic variety by Corollary A.5 (ii).
It remains to show that S c is proper. Since each h i is not identical zero , h −1 i ({0}) is a proper algebraic variety, and so is S c by Corollary A.4. This completes the proof.
We will now study matrix triples (E, A, B) ∈ ,n,m so that the polynomial matrix [s E − A, B] has "full" rank for each s = λ ∈ C. An important tool for these investigations is coprime polynomials and the characterization by the resultant.
· · · · · q 0 p n p n−1 · · q n · · q 1 p n · · · · · · · · p 0 q m−1 · · · · · · q m · · · · · · · · · · · · · p n q m The matrix above is called the Sylvester matrix of p(s) and q(s). The Sylvester matrix contains m columns with the coefficients of p and n columns with the coefficients of q, so that it is in R (n+m)×(m+n) . All other entries are zero. Note that the diagram shows the case n < m.
A well-known characterization of coprime polynomials in terms of the Sylvester matrix is given in the following lemma; for a proof see for example [7,Thm We are now in a position the characterize genericity of a set of pencils satisfying a rank condition in the complex plane.  E − A, B]) .

Proposition B.8 The set
Since ¬ d = = n + m and d ≤ min { , n + m}, we conclude d < max { , n + m}, and therefore r ≥ 2. Now, we are in a position to show the following implications for any (E, A, B) ∈ ,n,m and any λ ∈ C: Therefore, S := (E, A, B) ∈ ,n,m (9) holds ⊆ S.
Step 2 Define N := (2n + m) and, for any i ∈ r , the maximal degree of the image of ,n,m under M i as  (11) and, by the definition of α i , we find that M α i i = 0 for all i ∈ r . Choose k ∈ r so that α k = max α i i ∈ r > 0.
If we consider stabilizability of systems described by differential-algebraic equation, then we are interested in the rank of the matrix [λE − A, B] for λ belonging to the closed right half plane only.

Proof
⇒ Note that S = ∅ if, and only if, d ≤ min { , n + m}. Assume d = = n + m. By the well-known Hurwitz criterion (see [8, pp. 339]), we find that a polynomial whose coefficients do not have the same sign has at least one zero in the closed right half plane. Consider the set of (E, A, B) ∈ (n+m),n,m so that the coefficients of det[s E − A, B] do not have the same sign. Then, this set is nonempty, open and is included in S c . Openness yields that S c has a positive Lebesgue measure and thus S is not generic Corollary A.5 (i).
⇐ This is a consequence of Proposition B.8.